Analysis & PDE

  • Anal. PDE
  • Volume 5, Number 4 (2012), 777-829.

Smooth type II blow-up solutions to the four-dimensional energy-critical wave equation

Matthieu Hillairet and Pierre Raphaël

Full-text: Open access

Abstract

We exhibit C type II blow-up solutions to the focusing energy-critical wave equation in dimension N=4. These solutions admit near blow-up time a decomposition

u ( t , x ) = 1 λ ( N 2 ) 2 ( t ) ( Q + ε ( t ) ) ( x λ ( t ) ) , w i t h ε ( t ) , t ε ( t ) 1 × L 2 1 ,

where Q is the extremizing profile of the Sobolev embedding 1L2, and a blow-up speed

λ ( t ) = ( T t ) e | log ( T t ) | ( 1 + o ( 1 ) )  as  t T .

Article information

Source
Anal. PDE, Volume 5, Number 4 (2012), 777-829.

Dates
Received: 8 October 2010
Accepted: 13 May 2011
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731245

Digital Object Identifier
doi:10.2140/apde.2012.5.777

Mathematical Reviews number (MathSciNet)
MR3006642

Zentralblatt MATH identifier
1329.35207

Subjects
Primary: 35Q51: Soliton-like equations [See also 37K40]

Keywords
wave equation blow-up

Citation

Hillairet, Matthieu; Raphaël, Pierre. Smooth type II blow-up solutions to the four-dimensional energy-critical wave equation. Anal. PDE 5 (2012), no. 4, 777--829. doi:10.2140/apde.2012.5.777. https://projecteuclid.org/euclid.apde/1513731245


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References

  • S. Alinhac, Blowup for nonlinear hyperbolic equations, Progress in nonlinear differential equations and their applications 17, Birkhäuser, Boston, MA, 1995.
  • J. B. van den Berg, J. Hulshof, and J. R. King, “Formal asymptotics of bubbling in the harmonic map heat flow”, SIAM J. Appl. Math. 63:5 (2003), 1682–1717.
  • P. Bizoń, T. Chmaj, and Z. Tabor, “Formation of singularities for equivariant $(2+1)$-dimensional wave maps into the 2-sphere”, Nonlinearity 14:5 (2001), 1041–1053.
  • D. Christodoulou and A. S. Tahvildar-Zadeh, “On the regularity of spherically symmetric wave maps”, Comm. Pure Appl. Math. 46:7 (1993), 1041–1091.
  • R. Cote, Y. Martel, and F. Merle, “Construction of multisolitons solutions for the $L^2$-supercritical gKdV and NLS equations”, preprint, 2009.
  • T. Duyckaerts, C. Kenig, and F. Merle, “Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation”, J. Eur. Math. Soc. $($JEMS$)$ 13:3 (2011), 533–599.
  • T. Duyckaerts, C. Kenig, and F. Merle, “Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case”, J. Eur. Math. Soc. $($JEMS$)$ 14:5 (2012), 1389–1454.
  • G. Fibich, F. Merle, and P. Raphaël, “Proof of a spectral property related to the singularity formation for the $L\sp 2$ critical nonlinear Schrödinger equation”, Phys. D 220:1 (2006), 1–13.
  • M. G. Grillakis, “Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity”, Ann. of Math. $(2)$ 132:3 (1990), 485–509.
  • K. J örgens, “Das Anfangswertproblem im Grossen für eine Klasse nichtlinearer Wellengleichungen”, Math. Z. 77 (1961), 295–308.
  • P. Karageorgis and W. A. Strauss, “Instability of steady states for nonlinear wave and heat equations”, J. Differential Equations 241:1 (2007), 184–205.
  • O. Kavian and F. B. Weissler, “Finite energy self-similar solutions of a nonlinear wave equation”, Comm. Partial Differential Equations 15:10 (1990), 1381–1420.
  • C. E. Kenig and F. Merle, “Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation”, Acta Math. 201:2 (2008), 147–212.
  • J. Krieger and W. Schlag, “On the focusing critical semi-linear wave equation”, Amer. J. Math. 129:3 (2007), 843–913.
  • J. Krieger and W. Schlag, “Non-generic blow-up solutions for the critical focusing NLS in 1-D”, J. Eur. Math. Soc. $($JEMS$)$ 11:1 (2009), 1–125.
  • J. Krieger, W. Schlag, and D. Tataru, “Renormalization and blow up for charge one equivariant critical wave maps”, Invent. Math. 171:3 (2008), 543–615.
  • J. Krieger, W. Schlag, and D. Tataru, “Renormalization and blow up for the critical Yang–Mills problem”, Adv. Math. 221:5 (2009), 1445–1521.
  • J. Krieger, W. Schlag, and D. Tataru, “Slow blow-up solutions for the $H\sp 1(\mathbb{R}\sp 3)$ critical focusing semilinear wave equation”, Duke Math. J. 147:1 (2009), 1–53.
  • H. A. Levine, “Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu\sb{tt}=-Au+{\mathcal{F}}(u)$”, Trans. Amer. Math. Soc. 192 (1974), 1–21.
  • Y. Martel, “Asymptotic $N$-soliton-like solutions of the subcritical and critical generalized Korteweg–de Vries equations”, Amer. J. Math. 127:5 (2005), 1103–1140.
  • Y. Martel and F. Merle, “Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation”, Ann. of Math. $(2)$ 155:1 (2002), 235–280.
  • J. L. Marzuola and G. Simpson, “Spectral analysis for matrix Hamiltonian operators”, Nonlinearity 24:2 (2011), 389–429.
  • F. Merle and P. Raphael, “Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation”, Geom. Funct. Anal. 13:3 (2003), 591–642.
  • F. Merle and P. Raphael, “On universality of blow-up profile for $L\sp 2$ critical nonlinear Schrödinger equation”, Invent. Math. 156:3 (2004), 565–672.
  • F. Merle and P. Raphael, “The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation”, Ann. of Math. $(2)$ 161:1 (2005), 157–222.
  • F. Merle and P. Raphael, “Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation”, Comm. Math. Phys. 253:3 (2005), 675–704.
  • F. Merle and P. Raphael, “On a sharp lower bound on the blow-up rate for the $L\sp 2$ critical nonlinear Schrödinger equation”, J. Amer. Math. Soc. 19:1 (2006), 37–90.
  • F. Merle and H. Zaag, “Determination of the blow-up rate for the semilinear wave equation”, Amer. J. Math. 125:5 (2003), 1147–1164.
  • F. Merle and H. Zaag, “Openness of the set of non-characteristic points and regularity of the blow-up curve for the 1 D semilinear wave equation”, Comm. Math. Phys. 282:1 (2008), 55–86.
  • P. Raphael, “Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation”, Math. Ann. 331:3 (2005), 577–609.
  • P. Raphaël and I. Rodnianski, “Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang–Mills problems”, Publ. Math. Inst. Hautes Études Sci. 115:1 (2012), 1–122.
  • P. Raphaël and J. Szeftel, “Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS”, J. Amer. Math. Soc. 24:2 (2011), 471–546.
  • M. Reed and B. Simon, Methods of modern mathematical physics, vol. 4: Analysis of operators, Academic Press, New York, 1978.
  • I. Rodnianski and J. Sterbenz, “On the formation of singularities in the critical ${\rm O}(3)$ $\sigma$-model”, Ann. of Math. $(2)$ 172:1 (2010), 187–242.
  • J. Shatah and A. S. Tahvildar-Zadeh, “On the Cauchy problem for equivariant wave maps”, Comm. Pure Appl. Math. 47:5 (1994), 719–754.
  • C. D. Sogge, Lectures on nonlinear wave equations, Monographs in Analysis 2, International Press, Boston, MA, 1995.
  • W. A. Strauss, Nonlinear wave equations, CBMS Regional Conference Series in Mathematics 73, American Mathematical Society, Providence, RI, 1989.
  • M. Struwe, “Globally regular solutions to the $u\sp 5$ Klein–Gordon equation”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. $(4)$ 15:3 (1988), 495–513.
  • M. Struwe, “Equivariant wave maps in two space dimensions”, Comm. Pure Appl. Math. 56:7 (2003), 815–823.